Aeroelastic Dynamic Response and Control of an Airfoil Section … · 2013. 6. 17. · Aeroelastic Dynamic Response and Control of an Airfoil Section with Control Surface Nonlinearities

Aeroelastic Dynamic Response and Control of an Airfoil Section

with Control Surface Nonlinearities

Daochun Li1, 2, Shijun Guo1, Jinwu Xiang2

1Aerospace Engineering, Cranfield University, Cranfield, Beds, MK43 0AL, UK2Beijing University of Aeronautics and Astronautics, Beijing 100083, P.R. China

Abstract Nonlinearities in aircraft mechanisms are inevitable, especially in the control

system. It’s necessary to investigate the effects of them on the dynamic response and control

performance of aeroelastic system. In this paper, based on the state-dependent Riccati

equation method, a state feedback suboptimal control law is derived for aeroelastic response

and flutter suppression of a three degree-of-freedom typical airfoil section. With the control

law designed, nonlinear effects of freeplay in the control surface and time delay between the

control input and actuator are investigated by numerical approach. A cubic nonlinearity in

pitch degree is adopted to prevent the aeroelastic responses from divergence when the flow

velocity exceeds the critical flutter speed. For the system with a freeplay, the responses of

both open- and closed-loop systems are determined with Runge-Kutta algorithm in

conjunction with Henon’s method. This method is used to locate the switching points

accurately and efficiently as the system moves from one subdomain into another. The

simulation results show that the freeplay leads to a forward phase response and a slight

increase of flutter speed of the closed-loop system. The effect of freeplay on the aeroelastic

response decreases as the flow velocity increases. The time delay between the control input

and actuator may impair control performance and cause high-frequency motion and

quasi-periodic vibration.

Keywords: aeroelasticity; freeplay nonlinearity; time delay; limit cycle oscillation; state-dependent Riccati

equation

li2106TextBoxJournal of Sound and Vibration, Volume 329, Issue 22, 25 October 2010, Pages 4756-4771

2

Nomenclature

a = non-dimensional distance from airfoil mid-chord to elastic axis

b = airfoil semi-chord

C(k) = generalized Theodorsen function

c = non-dimensional distance from airfoil mid-chord to the control surface hinge line

ci = coefficients of Wagner’s function

h = plunge displacement

J = performance index of optimal problem

k = reduced frequency

L = aerodynamic lift

M , M = aerodynamic moment of wing-aileron and of aileron

m = mass of wing-aileron (per unit span)

tm = total mass of wing-aileron and support blocks (per unit span)

Q = state coefficient matrix in performance index

r = control input coefficient in performance index

r = radius of gyration of wing-aileron

r = reduced radius of gyration of aileron

t = time

at = time when the control is turned on

U = free stream velocity

ax = vector of augmented variables

x = non-dimensional distance from airfoil elastic axis to center of mass

x = non-dimensional distance from aileron hinge line to center of mass

= pitch angle about the elastic axis

= aileron displacement about the hinge line

c = control input

3

= freeplay region

= cubic coefficient in pitch

= density of air

= damping ratio

= time delay

= uncoupled natural frequency

1. Introduction

Due to various nonlinearities, aeroelastic systems exhibit a variety of phenomena such as limit cycle

oscillation and chaotic vibration [1-3]. Flutter instability can jeopardize aircraft structure and its performance.

A great deal of research activity devoted to flutter control of aeroelastic system has been accomplished.

Kurdila et al. [4] gave an extensive review of nonlinear control methods for high-energy limit-cycle

oscillations. Mukhopadhyay [5] presented an historical perspective on analysis and control of aeroelastic

responses. In recent years, a large number of control strategies have been developed for the flutter

suppression [6-14, 16, 17], such as adaptive decoupled fuzzy sliding-mode control [6], and tensor-product

model-based control [7]. In Ref. 7, parameter-varying state-space model was transformed into the tensor-

product model whereupon linear matrix inequality techniques in the parallel distributed compensation design

framework can be executed to define controller. As an extension of Ref. 7, an observer was derived via

LMI-based design to estimate the practically unmeasurable state values from the output values [8]. A

multiple-input and multiple-output adaptive control law was designed via both leading and trailing edge

control surfaces [9]. For the two-degree-of-freedom aeroelastic system with uncertainties, many effective

adaptive control laws were designed by Singh et al. [10-14]. Mracek et al. [15] carried out a control design of

the nonlinear benchmark problem using the state-dependent Riccati equation (SDRE) method. Then SDRE

control technique was developed to design suboptimal control laws for nonlinear aeroelastic systems [16-17].

Time delays in control loops are inevitable because of the dynamics involved in the actuators, sensors,

and controllers [18], and are prevalent when digital controllers, analogue anti-aliasing and reconstruction

filters, and hydraulic actuators were used [19]. Time delay feedback control has received much attention

recent years [20]. Chaotic motions of a two-dimensional airfoil are controlled by the application of the

time-delayed continuous feedback method of Pyragas in Ref. [21]. Four control strategies are implemented

with plunging displacement, plunging velocity and pitching angle, pitching velocity. It showed that the

4

feedback control signal derived from the pitching variables was more effective in controlling the chaotic

motion of the airfoil. Time delay effects on linear/nonlinear feedback control of simple aeroelastic system

were presented by Marzocca et al. [22]. In Yu et al. [23], the problem of implications of time delay feedback

control of a two-dimensional supersonic lifting surface on flutter boundary is addressed. And they pointed

out that we should apply both linear and nonlinear controls with small time delay, and avoid using plunging

displacement controls. Zhao [18] presented a systematic study on aeroelastic stability of a two-dimensional

airfoil with a single or multiple time delays in the feed back control loops. On the other side, time delay

between the control input and actuators is unavoidable, and should be taken into account in the process of the

control of aeroelastic system. As indicated in Ref. [22], the actuators may input energy at the moment when

the controlled system does not need it. These delays can be very detrimental in the sense of impairing the

control performance and can even cause irregular motions, producing instability. So it is of interest to

investigate the effect of time delay on an aeroelastic system.

A state-space linear model with control input of a typical three degrees-of-freedom airfoil section was

developed by Edwards et al. [24]. Conner et al. [25] successfully adapted the model to investigate the effect

of structural freeplay on an open-loop system response in numerical and experimental approach. In previous

research, the flutter control was studied without considering nonlinearity such as freeplay in the control

surface [6-17]. In the current paper, freeplay nonlinearity in the control surface has been considered in the

design of a state feedback control law for flutter suppression. With the control law designed, the effect of

control surface freeplay on the dynamic response and flutter suppression have been investigated. In addition,

the effect of time delay between the actuator control input and the control surface action is also investigated.

The rest of this paper is organized as follows: Section 2 presents the aeroelastic equation and control

problem. A feedback control law is designed in Section 3. Section 4 shows simulation results and discussion.

Conclusions are presented in Section 5.

2. Equations of Motion and Control Problem

A typical airfoil section with a trailing edge control surface is normally simplified and modelled as a three

degrees-of-freedom system as illustrated in Fig. 1 for aeroelastic analysis [25]. The three degrees of freedom

of the model include plunge h, pitch , and control surface angle . The elastic axis is located at a distance

ab from the mid-chord. The airfoil mass center is located at a distance x b from the elastic axis, where b is

5

the semi-chord of the airfoil section. Both distances are positive when measured towards the airfoil trailing

edge. There is a distance cb from the hinge line of the control surface, and a distance x b from the mass

center of the control surface to the mid-chord.

The non-dimensional governing equations of motion for the airfoil aeroelastic model are given by

2 2 2 2 2

2 2 2 2 2

2

/( )

/( )

/ /( )t h

r r c a x x h r F M mb

r c a x r x h r G M mb

x x m m h h L mb

(1)

The parameters in Eq. (1) are defined in the nomenclature. In order to keep the system response in limited

cycle beyond flutter speed, a cubic nonlinear spring in pitch is considered for F() to be

3F (2)

The control surface moment rotation relationship considering freeplay nonlinearity in the control surface

is illustrated in Fig. 2 and expressed as

( ) 0G

(3)

A structure damping matrix is created in this model according to Ref. [25, 26]. The eigenvalue i , the

natural frequency i i , and eigenvector matrix Λ are determined from the left-hand side structure

components of Eq. (1). Then the system modal mass matrix Tmod sM M and the modal damping

matrix Bmod can be obtained

modB

2 0 0

0 2 0

0 0 2 h h h

m

m

m

(4)

where im are the values at the diagonal entries of modM , and i are the measured damping ratios. From

the matrix Bmod, the structure damping matrix can be obtained as 1 1

s modTB B

.

The unsteady aerodynamic force and moments in incompressible flow are given by [27]

6

2 2 2 24 10

21 8 4 11 7 1

210 11

(1/ 2 ) (1/ 8 ) ( )

( ) (1/ 2) ( )

2 ( 1/ 2) ( ) (1/ 2 ) 1/ (1/ 2 )

M b a Ub b a T T U

T T c a T T Ub T c a T b a bh

Ub a C k U h b a T U b T

(5)

2 29 1 4 13

2 25 4 10 4 11 3 1

212 10 11

2 ( 1/ 2) 2

(1/ ) ( ) (1/ 2 ) (1/ )

( ) (1/ 2 ) 1/ (1/ 2 )

M b T T T a Ub T b

U T T T Ub T T T b T bh

Ub T C k U h b a T U b T

(6)

24 1

10 11

( )

2 ( ) (1/ 2 ) 1/ (1/ 2 )

L b U h ba UT T b

UbC k U h b a T U b T

(7)

The Theodorsen constants Ti, i =1, 2,…13, are given in Appendix A. The aerodynamic force and moments

in Eq. (5) to Eq. (7) are dependent on reduced frequency k. So Eq. (1) is restricted to simple harmonic

oscillation.

Aerodynamics in Eq. (5) to Eq. (7) is dependent on Theodorsen’s function, C(k), where k is the

nondimensional reduced frequency of harmonic oscillation. So the aerodynamics is restricted to simple

harmonic motion. In order to simulate arbitrary motion of the airfoil, the loading associated with

Theodorsen’s function C(k)f(t) is replaced by the Duhamel formulation in the time domain

0

0cf

L C k f t f d

(8)

where,

10 11(1/ 2 ) 1/ (1/ 2 )f t U h b a T U b T (9)

and is Wagner function. In this paper, convenient approximation of Sears is used as

2 40 1 3c cc c e c e (10)

The coefficients in Eq. (10) are 1 0.165c , 2 0.0455c , 3 0.335c , and 4 0.3c .

In order to simplify the Theodorsen function, rewrite the Duhamel integral using integration by parts,

00cL f f d

(11)

If we follow the state space method used by Lee et al. [28-29], six augmented states will be needed. Here, the

Padé approximant method is used to represent the integral term as a second order ordinary differential

equation as follow,

7

0 1 3 2 4 1 3 1 2 3 4cL c c c f t c c c c x c c c c x (12)

After two augmented variables are introduced: 1ax x , 2ax x , Eq. (1) can be expressed in a matrix

form [30],

s NC s NC 2 s NC 1 3 a1/ 2 1/ 2 0M M x B B RS x K K RS x RS x (13)

where T

h b x / , and 1 2axT

a ax x . Further details of the matrices and the additional vector

terms are given in Appendix B.

After introducing a variable vectorTT T T

a X x x x ,

8X R , Eq. (13) can be rewritten in a

state-space form

cX A X X B (14)

where c is the command input. The definition of matrix A X is given in Appendix C. Matrix B is

given by

3 1

1

2 1

0

0t

B M G (15)

where G 2 20 0 r .

If a time delay exists between the control input and actuator, Eq. (14) becomes

cβ t-τ X A X X B (16)

3. Control Law Design and Numerical Integration

In this section, a nonlinear flutter control law based on the state-dependent Riccati equation method

[15-17] is designed. Consider the optimal infinite-horizon regulator problem, the performance index J is to

be minimized subject to the system expressed by Eq. (14).

20

1

2T

cJ r dtX Q X X

(17)

where Q X is a positive definite symmetric matrix and 0r for 8X R .

For the system modeled in Eq. (14), the controllability matrix is given by

2 8, ( ) , ( ) , ( ) B A X B A X B A X B (18)

8

In order to obtain the suboptimal solution of the preceding problem, we solve the state-dependent Riccati

equation given by

1 0T T A X P X P X A X P X BR B P X Q X (19)

for a symmetric positive definite matrix P X , the nonlinear feedback control law is given by

1 T c X R B P X X (20)

Substituting Eq. (20) into Eq. (14), the closed-loop system is obtained as follow

X A X X (21)

where the closed-loop system matrix is given by

1 T A X A X BR B P X (22)

Due to the freeplay nonlinearity in control surface, the system is divided into three distinct subdomains.

In the numerical integration process, a key problem is to locate the switching point at which the system

moves from one subdomain into the next. Numerical instability may occur unless the switching points are

located accurately. Based on the work done by Henon [31] and Conner et al. [32], the Henon’s method is

developed to solve the aeroelastic control problem.

Equation (21) is then rewritten in the following form

1 1 1

2 2 2

8 8 8

x x f

x x fd

dt

x x f

X

XA X

X

(23)

Divide each of the state equations in Eq. (23) by 2 2dx /dt=f X , and replace the second equation by

2 =1// X2dt dx f . Then the closed-loop equation of motion can be rearranged as

1

1 1 2

2

2

8 8 2

x f / f

t / fd

dx

x f / f

X X

X

X X

(24)

where the control surface displacement becomes an independent variable, and the time becomes a dependent

variable.

By using the Runge-Kutta method, Eq. (23) is integrated forward in time until a change in the

subdomain is detected. Then, the altered system (24) can be integrated using the distance from the current

9

control surface position to the boundary that has been crossed as step size. The result of Henon integration

can be used as the initial condition to integrate the original state equation in a new subdomain until the next

boundary is crossed.

4. Results and Discussion

Numerical simulation results are presented in this section. A small time step of 0.0001s is set in the

process of numerical integration. The values for the system parameters are taken from Ref. [25] and listed as

follows: a = -0.5, b = 0.127m, c = 0.5, s = 0.52m, mT = 1.2895 kg, I= 0.01347 kg.m2, I= 0.0003264 kg.m

2,

x= 0.434, x= 0.01996, kh = 2818.8 N/m, S= 0.08587 kg.m, S= 0.00395 kg.m, = 1.225 kg/m3,

1 0.01626 , 2 0.0115 , 3 0.0113 . The nonlinear parameter in pitch stiffness is = 3.

4.1 Open-loop system response with freeplay nonlinearity

Starting from the open-loop stability study of the aeroelastic system, the system response keeps stable

until the velocity reaches U = 23.96 m/s when no freeplay exists. However, the system with freeplay exhibits

limit cycle oscillation at about U = 6.0 m/s, which is relative to the initial conditions used in the simulation.

To study the effect of control surface freeplay nonlinearity on the open-loop system response, the two values

of freeplay = 1.0o and = 2.0o were chosen in this case to compare with the = 0o non-freeplay system.

Flow velocity below the non-freeplay system flutter speed With the flow velocity U = 5.0 m/s, both

freeplay and non-freeplay systems are stable. From Fig. 3, it can be observed that the freeplay system

response needs more time to converge. In addition, the response amplitude is obviously bigger than the

non-freeplay system. At flow velocity U = 10.0 m/s, the freeplay system response become limit cycle

oscillation while the non-freeplay system remains stable as shown in Fig. 4. The control surface response

amplitude is slightly larger than the freeplay magnitude. However the freeplay magnitude has little influence

on the critical flutter velocity of the system. The system pitch and plunge amplitudes with = 2.0o are twice

larger than the amplitudes with = 1.0o.

Flow velocity above the non-freeplay system flutter speed At the flow velocity U = 25.0 m/s, Fig. 5

shows that the aeroelastic response of non-freeplay system becomes limit cycle oscillation. The response of

freeplay system is also under limited amplitude, but not divergent, although the flow velocity is beyond the

non-freeplay critical flutter speed. It indicates that the cubic nonlinearity in pitch may transform the freeplay

system response from divergent motion to a limit cycle oscillation with limited amplitude. The freeplay

10

system responses in pitch and plunge are similar to harmonic motions with slightly larger amplitudes than the

non-freeplay system responses. However, there is an obvious difference between the freeplay and non-

freeplay control surface deflections as shown in Fig. 5.b. Frequency analysis of the control surface responses

show that the first vibration mode component (6.0 Hz) is dominant in the non-freeplay flap response (Fig.

6.a), while the third vibration mode component (18.0 Hz) make more contribution when the freeplay exists in

the control surface (Fig. 6.b). If flow velocity is continuously increased up to U = 30.0 m/s, the third

vibration mode component (18.9 Hz) will take important role in both cases as shown in Fig. 6.c and 6.d. So

the freeplay system has almost the same motion as non-freeplay system except for the phase advance as

shown in Fig. 7.a.

As the existence of freeplay, the stiffness of the connection between main airfoil and control surface is

reduced. When the main airfoil vibrates, the control surface is motivated through the freeplay connection.

The lower connection stiffness will postpone the control surface response. But in the current case of

self-excited vibration under unsteady aerodynamics, the freeplay system is easier to be excited and more

sensitive to the aerodynamics. Figure 7.b shows the phase transformation process in the time history of

control surface deflection. It shows the non-freeplay system response is forward in phase at the beginning

until the unsteady aerodynamics become dominant in the vibration.

The analysis above indicates that the effect of freeplay on aeroelastic response is reducing as the

deflection amplitude of the control surface alone with the flow velocity increases. From Fig. 2, it is noted

that when the control surface deflection is small, the freeplay is dominant in the relationship between torque

and the deflection. However, if the control surface deflects with larger amplitude, the linear component will

become dominant. In other words, the freeplay nonlinearity has less effect and the system behaves more like

a non-freeplay system as the amplitude increases.

4.2 Closed-loop system response with freeplay nonlinearity

In the following closed-loop situation, state coefficient matrix Q in performance index is chose as a 8*8

unit matrix, and control input coefficient r = 100. To study the effect of control surface freeplay nonlinearity

on the closed-loop system response, the value = 2.0o was chosen in this case to compare with the = 0o

non-freeplay system. The simulation was first performed with the flow velocity U = 24.0 m/s, which is above

both freeplay and non-freeplay critical flutter speed. As shown in Fig. 8, after initial oscillatory transients the

plunge and pitch states converge to zero in about two seconds. There is no obvious difference between the

11

freeplay and non-freeplay system responses. At a higher flow velocity U = 32.0 m/s, a faster response time of

less than one second was obtained and shown in Fig. 9. The freeplay response maximum amplitude is

obviously higher. The freeplay states take more time to converge to zero after the transient oscillation, which

behaves like a response of an overdamping system. Given the initial condition of = 0.1 rad and the other

variables zero, the closed-loop system will keep stable until the flow velocity reaches 84.3 m/s. While the

freeplay system (= 2.0o) will show limit cycle oscillation when the flow velocity U > 86.7 m/s.

If the control input was triggered after the system had exhibited limit cycle oscillation, the closed-loop

critical flutter speed would drop down at a lower value as a bigger control input would be required to

suppress the limit cycle oscillation than the initial disturbance. For example, the critical flutter speed of the

system without freeplay was reduced from 84.3 m/s to 27.3 m/s. While the freeplay (= 2.0o) led to a slight

increase of the flutter speed to 27.6 m/s. This is because the freeplay makes the control surface hinge

stiffness decrease. So the control surface is easier to be actuated to follow the control input. A comparison of

the closed-loop responses at U = 27.4 m/s are shown in Fig. 10, where the controller is triggered at time t =

2s after the system exhibited limit cycle oscillation. It shows that the closed-loop response without freeplay

is in limit cycle oscillation with reduced amplitude, while the response with freeplay actually converges.

4.3 Closed-loop system response with time delay

In this section, the freeplay nonlinearity is ignored. Thus the standard 4-order Runge-Kutta algorithm, in

stead of the Henon’s method is used in the integrating process. The cubic nonlinearity in pitch is preserved to

prevent the system response from divergence. In this case, the simulation was conducted at the flow velocity

below and above the critical flutter speed.

System response below the flutter speed First a comparison was made between the open- and closed-loop

system responses at U =10.0 m/s without any time delay. From the results shown in Fig. 11, it can be

observed that the control law designed in section 3 makes the plunge of the closed-loop system converge

much faster than the open-loop system. There is no obvious difference in the pitch response. If a time delay

between the control input and actuator occurred at time t, the control input c(t-) would be derived from the

previous state X(t-) at time before the present state. The control input c(t-) would drive the system

based on the state X(t-) including deflection (t-), which would cause oscillations of the system state and

control input. It can be observed that this vibration was convergent in the beginning. When >0.3 ms, a

high-frequency vibration of small amplitude arose in the control input c (t), but the vibration is convergent.

12

Until ≥1.1ms, the vibration becomes divergent. Here, the control input was limited to a range of deflection

angle ±20o. The control input time histories of the close-loop system for three typical time delay sizes were

shown in Fig. 12. From the analysis results, it was also found that the time delay will produce an additional

motion in the system responses. Given =2.0ms as an example as shown in Fig. 13, a high-frequency

vibration with little amplitude occurred in company with the main vibration.

System response above the flutter speed When the flow velocity was above the system flutter speed, a

control law was designed to keep the system response convergent. If a time delay was set between the

control input and actuator however, the system response behaved differently. For example, the closed-loop

critical flutter speed decreased from U = 84.3 m/s to 47.0 m/s when a time delay of = 2.0 ms was set in the

system. The system responses at U = 60.0 m/s in terms of pitch angle and plunge displacement are shown in

Fig. 14. It is obvious that the pitch and plunge responses with the time delay does not converge.

If the time delay was further increased, the designed control law would completely invalid. The system

exhibit limit cycle oscillations even the flow velocity is much lower than open-loop critical flutter speed

(23.96 m/s). For example, with the time delay =10.0ms, two typical vibration styles are performed for U =

10.0 m/s and U = 25.0 m/s. At the flow velocity U = 10.0 m/s below the open-loop critical flutter speed, the

time history and phase diagram results of the system are shown in Fig. 15. The phase diagrams and Poincare

section indicate that the motion is quasi-periodic. At the flow velocity U = 25.0 m/s above the open-loop

critical flutter speed, the phase diagrams of pitch and plunge responses are shown in Fig. 16. The system

response becomes more complex than common quasi-periodic motion. But in fact, the Poincare sections in

circles indicate that the motion remains as quasi-periodic motion. The study results indicate that the time

delay between the control input and actuator may jeopardize the performance of a carefully designed control

law, and cause instability of the system.

Keep the flow speed U = 25.0 m/s, bifurcation diagrams of closed-loop system response as function of

time delay are given in Fig. 17. From Fig. 17.a, we can see that the system shows limit cycle oscillation after

a supercritical Hopf bifurcation at about =1.2 ms. However, the amplitudes of pitch and plunge responses

are quite small until the time delay is higher than 4.0 ms. Which means that a quite small time delay may

lead to system instability, but the control law designed is still effective to depress the flutter. With the time

delay increasing, closed-loop response amplitudes become large rapidly. At =10 ms, the pitch and plunge

amplitudes are almost the same with open-loop response amplitudes. And now the control law has not any

effectiveness for flutter control.

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5. Conclusions

Based on the state space model of a three degrees-of-freedom airfoil section with freeplay nonlinearity, a

suboptimal control law was designed by using the state-dependent Riccati equation method and applied for

dynamic response suppression in this paper. In the model, a cubic nonlinearity term was adopted to prevent

the aeroelastic response from divergence when the flow velocity is beyond the system critical flutter speed.

The effects of both control surface freeplay and time delay between the control input and actuator on the

aeroelastic responses have been investigated. The Henon’s method was employed to locate the switching

points in the procedure of Runge-Kutta numerical integration.

Because the linear component in the control surface response amplitude increases and becomes dominant,

the effect of freeplay on the system response reduces as the flow velocity increases. In addition, the freeplay

shows some beneficial effect on the aeroelastic stability of the closed-loop system. Due to the freeplay, the

system response has a forward phase than the non-freeplay system response. The system response is sensitive

to the time delay between the control input and actuator. The bifurcation diagram of system response as

function of time delay indicates that a supercritical Hopf bifurcation occurred with a small time delay, which

leads to high-frequency vibration. And with the time delay increasing, the system responses become

quasi-periodic motions. It should be noted that the effect of time delay on the system response depends upon

the control law designed in this paper. With alternative control laws, the system dynamic response might

exhibit different phenomena. In a more realistic model, the actuator stiffness and damping should be taken

into account in the controller design.

Acknowledgments

The authors would like to acknowledge the EC funding for the EU FP7 collaborative project SADE and

the National Nature Science Grant of P.R. China (No. 90916006).

Appendix A: Theodorsen Constants in Eq. (5) to Eq. (7)

2 2 11 1/ 3 1 2 cosT c c c

22 2 2 2 1 2 1

3 1/8 1 5 4 1/ 4 7 2 1 cos 1/8 cosT c c c c c c c c

2 14 1 cosT c c c

14

22 2 1 1

5 1 2 1 cos cosT c c c c c

2 2 17 1/8 7 2 1/8 cosT c c c c

2 2 18 1/ 3 1 2 1 cosT c c c c

3

29 41/ 2 1/ 3 1T c aT

2 110 1 cosT c c

1 211 cos 1 2 1 2T c c c c

2 112 1 2 cos 2 1T c c c c

13 7 11/ 2T T c a T

Appendix B: Definitions of Matrices Appearing in Eq. (13)

sM

2 2

2 2

t

r r c a x x

r c a x r x

x x M / M

sB 1

1

2 0 0

0 2 0

0 0 2

T

h h h

m

m

m

sK

2 2

2 2

2

/ 0 0

0 / 0

0 0 h

r F

r G

NCM

-

2 2 2 27 1

2 2 213 3 1

2 2 21

1/ 8

2 /

b a T c a T b ab

T b T b T bm

ab T b b

NCB

1 8 4 11

9 1 4 4 11

4

1/ 2 / 2 0

2 1/ 2 / 2 0

0

a Ub T T c a T T Ub

T T T a Ub T T Ubm

Ub UT b

15

NCK

24 10

25 4 10

0 0

0 / 0

0 0 0

T T U

T T T Um

R 122 1/ 2 / / 2 / T

U a m UT m U m

1S 10 / 0 U T U

2S 111/ 2 / 2 b a bT b

3S 2

2 4 1 3 1 2 3 4/ c c c c U b c c c c U .

Appendix C: Definitions of Matrices Appearing in Eq. (14)

A

3 3

1 1 1

1 2

0 0

t t t t t

I

M K M B M D

E E F

,

where, t s NCM M M , 21/ 2t s NCB B B RS , 11/ 2t s NCK K K RS , 3D RS ,

1E 10

0 0 0

/ / 0

U b UT b

, 2E 11

0 0 0

1/ 2 / 2 1

a T

, F 22 4 2 4

0 1

/ /

c c U b c c U b

.

References

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Figure Captions

Fig. 1. Schematic of airfoil section with a control surface.

Fig. 2. Freeplay nonlinearity.

Fig. 3. Time histories of open-loop aeroelastic systems at U = 5.0 m/s: a) pitch; b) flap; c) plunge.



Fig. 6. Frequency analysis of flap deflection for a) and b): at U = 25.0 m/s; c) and d): at U = 30.0 m/s.

Fig. 7. Time histories of open-loop aeroelastic systems at U = 30.0 m/s: a) stable; b) transient.

Fig. 8. Aeroelastic flutter suppression with and without freeplay at U = 24.0 m/s: a) pitch; b) plunge.



Fig. 11. Open-loop and closed-loop responses at U = 10.0 m/s, = 0.0 ms: a) pitch; b) plunge.

Fig. 12. Time histories of control input at U = 10.0 m/s and: a) = 0.5 ms; b) =1.0 ms; c) =1.1 ms.

Fig. 13. Closed-loop responses with and without time delay at U =10.0 m/s, =2.0 ms: a) pitch; b) plunge.

Fig. 14. Time histories at U = 60.0 m/s, =2.0 ms: a) pitch; b) plunge.

Fig. 15. Time histories a) pitch; b) plunge; phase diagrams c) pitch; d) plunge; and Poincare sections e) pitch;

f) plunge at U = 10.0 m/s, = 10.0 ms.

Fig. 16. Time histories a) pitch; b) plunge; phase diagrams c) pitch; d) plunge; and Poincare sections e) pitch;

f) plunge at U = 25.0 m/s, = 10.0 ms.

Fig. 17. Bifurcation diagrams of a) pitch; b) plunge response as function of time delay at U = 25.0 m/s.

19

Fig. 1. Schematic of airfoil section with a control surface.

20

Fig. 2. Freeplay nonlinearity.

21

a)

b)

c)

Fig. 3. Time histories of open-loop aeroelastic systems at U =5.0 m/s: a) pitch; b) flap; c) plunge.

22

a)

b)

c)


23

a)

b)

c)


24

a) b)

c) d)

Fig. 6. Frequency analysis of flap deflection for a) and b): at U = 25.0 m/s; c) and d): at U = 30.0 m/s.

25

a)

b)

Fig. 7. Time histories of open-loop aeroelastic systems at U = 30.0 m/s: a) stable; b) transient.

26

b)


27

a)

b)


28

a)

b)


29

a)

b)

Fig. 11. Open-loop and closed-loop responses at U = 10.0 m/s, = 0.0 ms: a) pitch; b) plunge.

30

a)

b)

c)

Fig. 12. Time histories of control input at U = 10.0 m/s and: a) =0.5 ms; b) =1.0 ms; c) =1.1 ms.

31

a)

b)

Fig. 13. Closed-loop responses with and without time delay at U =10.0 m/s, =2.0 ms: a) pitch; b) plunge.

32

a)

b)

Fig. 14. Time histories at U = 60.0 m/s, = 2.0 ms: a) pitch; b) plunge.

33

a) b)

c) d)

e) f)

Fig. 15. Time histories a) pitch; b) plunge; phase diagrams c) pitch; d) and plunge Poincare sections e) pitch;

f) plunge at U = 10.0 m/s, =10.0 ms.

34

a) b)

c) d)

e) f)

Fig. 16. Time histories a) pitch; b) plunge; phase diagrams c) pitch; d) plunge; and Poincare sections e)

pitch; f) plunge at U = 25.0 m/s, = 10.0 ms.

35

a)

b)

c)

Fig. 17. Bifurcation diagrams of a) , b) pitch and c) plunge response as function of time delay at U = 25.0

m/s.

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Aeroelastic Dynamic Response and Control of an Airfoil Section … · 2013. 6. 17. · Aeroelastic Dynamic Response and Control of an Airfoil Section with Control Surface Nonlinearities